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Is |x - 1| less than 1 ?
1). (x - 1)^2 less than and equal to 1
2). x^2 - 1 greater than 0
Answer: E
|x-1| less than 1 is only true when 0 less than x less than 1
From statement (1): (x-1)^2<=1
True when 0<=x<=2
If x=0.5, then |x-1| less than 1 is true
If x=2, then |x-1| less than 1 is not true
Hence insufficient
From statement (2): x^2>1 means x>1 and x<-1
True when x=1.5, but not when x=3
Hence insufficient
Statement (1) and (2) together: 1 is less than x is less than and equal to 2
Taking x=1.5 and x=2
Hence insufficient
Which of the following is always equal to sqrt (9 + x^2 - 6x)?
a) x - 3
b) 3 + x
c)|3 - x|
d)|3 + x|
e) 3 - x
Answer: C
sqrt (9+x^2-6x)
= sqrt( (3-x)^2 )
= |3-x|
What is the value of x?
1) (-x)^3 = -x^3
2) (-x)^2 = -x^2
Answer: B
From statement (1):
if x = 0 both sides are equal
if x = 1 both sides are again equal {(-1)^3 = -1 & -1^3 = 1}
=> x = 0 or x = 1
Hence insufficient
From statement (2):
x can only be zero because the square of a number other than zero cannot be negative
{(-1)^2 = 1 which is not equal to -(1)^2)}
=> from above it is sufficient to say that x = 0
Hence sufficient
Is |x - 1| less than 1 ?
(1). (x - 1) ^2 less than and equal to 1
(2). (x^2) - 1 greater than 0
Answer: E
From statement (1): (x - 1) ^2 less than and equal to 1
Now this is true only if 0 is less than and equal to x and x is less than and equal to 2. (We know that (x - 1) ^2 less than 1 is only true when 0 is less than x and x is less than 2)When we take x = 0.5, then |x-1| less than 1 is true
When we take x = 2, then |x-1| less than1 does not holds true
Hence insufficient
From statement (2): x^2 greater than 1 => x is less than -1 and x is greater than 1
When we take x = 1.5, then x^2 greater than 1 is true
When we take x = 3, then x^2 greater than 1 does not holds true
Hence insufficient
Taking both statements (1) and (2) together -- We have 1 is less than x and x is less than and equal to 2
When we take x = 1.5, then 1 is less than x and x is less than and equal to 2 is true
When we take x = 2, then 1 is less than x and x is less than and equal to 2 does not holds true Hence insufficient
